##
**A projective plane in \({\mathbb{R}}^ 4\) with three critical points is standard. Strongly invertible knots have property P.**
*(English)*
Zbl 0678.57003

The main result of this paper is that if one gets the unknot by attaching a band to the unknot, then the band is a trivial band with a half twist. The proof uses variations on the combinatorics of the intersection of planar surfaces developed by the second author [Invent. Math. 79, 125-141 (1985; Zbl 0559.57019) and ibid. 82, 37-55 (1985; Zbl 0576.57004)] and which have turned out to be very important elsewhere in 3-manifold topology - e.g. in the solution of the knot complement problem and in the cyclic surgery theorem. The theorems of the title are derived as consequences.

Reviewer: J.Hempel

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57R40 | Embeddings in differential topology |

57R70 | Critical points and critical submanifolds in differential topology |